Problem: Let $f(x) = 9x^{2}-9x-3$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )?
Solution: The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $9x^{2}-9x-3 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = 9, b = -9, c = -3$ $ x = \dfrac{+ 9 \pm \sqrt{(-9)^{2} - 4 \cdot 9 \cdot -3}}{2 \cdot 9}$ $ x = \dfrac{9 \pm \sqrt{189}}{18}$ $ x = \dfrac{9 \pm 3\sqrt{21}}{18}$ $x =\dfrac{3 \pm \sqrt{21}}{6}$